Trigonometry/Geometric Definitions of Trig Functions

In the previous section, we algebraically defined tangent as tan⁡θ=sin⁡θcos⁡θ{\displaystyle \displaystyle \tan \theta ={\frac {\sin \theta }{\cos \theta }}}, and this is the definition that we will use most in the future. It can, however, be helpful to understand the tangent function from a geometric perspective.

Geometrically defining tangent

A line is drawn at a tangent to the unit circle: (i.e. x=1{\displaystyle x=1}). Another line is drawn from the point on the radius of the circle where the given angle falls, through the origin(O), to a point (Q) on the drawn tangent. The ordinate (QP) of this point is called the tangent of the angle.

The slope of the line OQ =KCOC{\displaystyle {\frac {KC}{OC}}} and as we mentioned before

KC = sin(θ) , OC = cos(θ)

Hence , the Slope of the line OQ =sin⁡(θ)cos⁡(θ){\displaystyle {\frac {\sin(\theta )}{\cos(\theta )}}}

Any size angle, positive or negative, can be the input to sine or cosine — the result will be as if the largest multiple of 2π (or 360°) were subtracted from or added to the angle. The output of the two functions is limited by the absolute value of the radius of the unit circle, |1|{\displaystyle |1|}.

No such restrictions apply to the tangent, however, as can be seen in the diagram in the preceding section. The only restriction on the domain of tangent is that odd integer multiples of π2{\displaystyle {\frac {\pi }{2}}} are undefined, as a line parallel to the tangent will never intersect it.